More on random walks, electrical networks, and the harmonic k-server algorithm

Yair Bartal; Marek Chrobak; John Noga; Prabhakar Raghavan

doi:10.1016/S0020-0190(02)00287-9

More on random walks, electrical networks, and the harmonic k-server algorithm

Yair Bartal, Marek Chrobak^*, John Noga, Prabhakar Raghavan

^*Corresponding author for this work

The Rachel and Selim Benin School of Engineering and Computer Science

Research output: Contribution to journal › Article › peer-review

9 Scopus citations

Abstract

Techniques from electrical network theory have been used to establish various properties of random walks. We explore this connection further, by showing how the classical formulas for the determinant and cofactors of the admittance matrix, due to Maxwell and Kirchoff, yield upper bounds on the edge stretch factor of the harmonic random walk. For any complete, n-vertex graph with distances assigned to its edges, we show the upper bound of (n - 1)². If the distance function satisfies the triangle inequality, we give the upper bound of 12n(n - 1). Both bounds are tight. As a consequence, we obtain that the harmonic algorithm for the k server problem is 12k(k + 1)-competitive against the lazy adversary.

Original language	English
Pages (from-to)	271-276
Number of pages	6
Journal	Information Processing Letters
Volume	84
Issue number	5
DOIs	https://doi.org/10.1016/S0020-0190(02)00287-9
State	Published - 16 Dec 2002

Bibliographical note

Funding Information:
* Corresponding author. Research supported by NSF grant CCR-9988360 and cooperative grant KONTAKT-ME476/CCR-9988360-001 from MŠMT Cˇ R and NSF. E-mail address: [email protected] (M. Chrobak). 1 Research supported by NSF grant CCR-9988360.

Keywords

Analysis of algorithms
Electrical networks
Randomization
Theory of computation

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10.1016/S0020-0190(02)00287-9

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title = "More on random walks, electrical networks, and the harmonic k-server algorithm",

abstract = "Techniques from electrical network theory have been used to establish various properties of random walks. We explore this connection further, by showing how the classical formulas for the determinant and cofactors of the admittance matrix, due to Maxwell and Kirchoff, yield upper bounds on the edge stretch factor of the harmonic random walk. For any complete, n-vertex graph with distances assigned to its edges, we show the upper bound of (n - 1)2. If the distance function satisfies the triangle inequality, we give the upper bound of 12n(n - 1). Both bounds are tight. As a consequence, we obtain that the harmonic algorithm for the k server problem is 12k(k + 1)-competitive against the lazy adversary.",

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AU - Chrobak, Marek

AU - Noga, John

AU - Raghavan, Prabhakar

N1 - Funding Information: * Corresponding author. Research supported by NSF grant CCR-9988360 and cooperative grant KONTAKT-ME476/CCR-9988360-001 from MŠMT Cˇ R and NSF. E-mail address: [email protected] (M. Chrobak). 1 Research supported by NSF grant CCR-9988360.

PY - 2002/12/16

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N2 - Techniques from electrical network theory have been used to establish various properties of random walks. We explore this connection further, by showing how the classical formulas for the determinant and cofactors of the admittance matrix, due to Maxwell and Kirchoff, yield upper bounds on the edge stretch factor of the harmonic random walk. For any complete, n-vertex graph with distances assigned to its edges, we show the upper bound of (n - 1)2. If the distance function satisfies the triangle inequality, we give the upper bound of 12n(n - 1). Both bounds are tight. As a consequence, we obtain that the harmonic algorithm for the k server problem is 12k(k + 1)-competitive against the lazy adversary.

AB - Techniques from electrical network theory have been used to establish various properties of random walks. We explore this connection further, by showing how the classical formulas for the determinant and cofactors of the admittance matrix, due to Maxwell and Kirchoff, yield upper bounds on the edge stretch factor of the harmonic random walk. For any complete, n-vertex graph with distances assigned to its edges, we show the upper bound of (n - 1)2. If the distance function satisfies the triangle inequality, we give the upper bound of 12n(n - 1). Both bounds are tight. As a consequence, we obtain that the harmonic algorithm for the k server problem is 12k(k + 1)-competitive against the lazy adversary.

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KW - Randomization

KW - Theory of computation

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More on random walks, electrical networks, and the harmonic k-server algorithm

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